Fractions define the relationship between parts and wholes (objects, groups of objects, or quantities).

A set is a collection of individual objects. Each object is an equal part of the whole.

A fraction always represents a part of something. This “something” is called the whole or 1.

The parts into which the whole are divided must be equal.

The number above the fraction bar is the numerator. It is the number of parts.

The number below the fraction bar is the denominator. It is the type of parts or the total number of parts that make up 1 whole.

An infinite number of different fractions can represent a single location on the number line. For example, , , , , etc. represent a single location on the number line.

The fraction bar is also called a “vinculum.” A vinculum is a horizontal line to indicate that multiple terms are considered as a single term.

Unit fractions are formed by dividing a whole (region, set, or distance on a number line) into equal parts1.

If the whole is divided into B equal parts, then the amount formed by 1 of those parts is of the whole. In other words, B copies of the amount of the whole are joined together make the whole. is the unit fraction2.

1 whole consists of B pieces of size .

Where A and B are counting numbers, the fraction of the whole is the amount formed by A parts (or copies of parts), each of which is of the whole2.

Using the English fraction term, is read: “four-sixths.” The number of parts being described is said first, “four.” Then, the name of the parts in the whole is said, “sixths.” 6 parts are involved, so each part is of the whole1.

can also be described as “out of 6 parts, select 4.” The number of the parts in the whole is indicated first, and then the number of parts is stated1.

The multiplicative identity property, also called the identity property of 1 or identity property of multiplication, states that the product of any number and 1 is the number itself.

1 is the multiplicative identity. 1 can be represented as , , , etc.

Because whole numbers are a subset of rational numbers, which include fractions, the multiplicative identify property applies when the original number is a fraction and when 1 is written in fractional form.

National Council of Teachers of Mathematics. (2009). Focus in grade 3: Teaching with curriculum focal points. Reston, VA: Author.